The n-Category Café

Our goal was to describe classical thermodynamics, classical statistical mechanics and quantum statistical mechanics in a unified framework based on entropy maximization. This framework can also handle ‘generalized probabilistic theories’ of the sort studied in quantum foundations—that is, theories like quantum mechanics, but more general.

To unify all these theories, we define a ‘thermostatic system’ to be any convex space $X$ of ‘states’ together with a concave function

$$S \colon X \to [-\infty, \infty]$$

assigning to each state an ‘entropy’.

Whenever several such systems are combined and allowed to come to equilibrium, the new equilibrium state maximizes the total entropy subject to constraints. We explain how to express this idea using an operad. Intuitively speaking, the operad we construct has as operations all possible ways of combining thermostatic systems. For example, there is an operation that combines two gases in such a way that they can exchange energy and volume, but not particles — and another operation that lets them exchange only particles, and so on.

It is crucial to use a sufficiently general concept of ‘convex space’, which need not be a convex subset of a vector space. Luckily there has been a lot of work on this, so we can just grab a good definition off the shelf:

Definition. A convex space is a set $X$ with an operation $c_\lambda \colon X \times X \to X$ for each $\lambda \in [0, 1]$ such that the following identities hold:

1) $c_1(x, y) = x$

2) $c_\lambda(x, x) = x$

3) $c_\lambda(x, y) = c_{1-\lambda}(y, x)$

4) $c_\lambda(c_\mu(x, y) , z) = c_{\lambda'}(x, c_{\mu'}(y, z))$ for all $0 \le \lambda, \mu, \lambda', \mu' \le 1$ satisfying $\lambda\mu = \lambda'$ and $1-\lambda = (1-\lambda')(1-\mu')$.

To understand these axioms, especially the last, you need to check that any vector space is a convex space with

$c_\lambda(x, y) = \lambda x + (1-\lambda)y$

So, these operations $c_\lambda$ describe ‘convex linear combinations’.

Indeed, any subset of a vector space closed under convex linear combinations is a convex space! But there are other examples too.

In 1949, the famous mathematician Marshall Stone invented ‘barycentric algebras’. These are convex spaces satisfying one extra axiom: the cancellation axiom, which says that whenever $\lambda \ne 0,$

$c_\lambda(x,y) = c_\lambda(x',y) \implies x = x'$

He proved that any barycentric algebra is isomorphic to a convex subset of a vector space. Later Walter Neumann noted that a convex space, defined as above, is isomorphic to a convex subset of a vector space if and only if the cancellation axiom holds.

Dropping the cancellation axiom has convenient formal consequences, since the resulting more general convex spaces can then be defined as algebras of a finitary commutative monad, giving the category of convex spaces very good properties.

But dropping this axiom is no mere formal nicety. In our definition of ‘thermostatic system’, we need the set of possible values of entropy to be a convex space. One obvious candidate is the set $[0,\infty).$ However, for a well-behaved formalism based on entropy maximization, we want the supremum of any set of entropies to be well-defined. This forces us to consider the larger set $[0,\infty],$ which does not obey the cancellation axiom.

But even that is not good enough! In thermodynamics you often read about the ‘heat bath’, an idealized system that can absorb or emit an arbitrarily large amount of energy while keeping a fixed temperature. We want to treat the ‘heat bath’ as a thermostatic system on an equal footing with any other. To do this, we need to allow consider negative entropies—not because the heat bath can have negative entropy, but because it acts as an infinite reservoir of entropy, and the change in entropy from its default state can be positive or negative.

This suggests letting entropies take values in the convex space $\mathbb{R}.$ But then the requirement that any set of entropies have a supremum (including empty and unbounded sets) forces us to use the larger convex space $[-\infty,\infty].$

This does not obey the cancellation axiom, so there is no way to think of it as a convex subset of a vector space. In fact, it’s not even immediately obvious how to make it into a convex space at all! After all, what do you get when you take a nontrivial convex linear combination of $\infty$ and $-\infty?$ You’ll have to read our paper for the answer, and the justification.

We then define a thermostatic system to be a convex set $X$ together with a concave function

$$S \colon X \to [-\infty, \infty]$$

where concave means that

$$S(c_\lambda(x,y)) \ge c_\lambda(S(x), S(y))$$

We give lots of examples from classical thermodynamics, classical and quantum statistical mechanics, and beyond—including our friend the ‘heat bath’.

For example, suppose $X$ is the set of probability distributions on an $n$-element set, and suppose $S \colon X \to [-\infty, \infty]$ is the Shannon entropy

$$\displaystyle{ S(p) = - \sum_{i = 1}^n p_i \log p_i }$$

Then given two probability distributions $p$ and $q,$ we have

$$S(\lambda p + (1-\lambda q)) \ge \lambda S(p) + (1-\lambda) S(q)$$

for all $\lambda \in [0,1].$ So this entropy function is convex, and $S \colon X \to [-\infty, \infty]$ defines a thermostatic system. But in this example the entropy only takes nonnegative values, and is never infinite, so you need to look at other examples to see why we want to let entropy take values in $[-\infty,\infty].$

After looking at examples of thermostatic systems, we define an operad whose operations are convex-linear relations from a product of convex spaces to a single convex space. And then we prove that thermostatic systems give an algebra for this operad: that is, we can really stick together thermostatic systems in all these ways. The trick is computing the entropy function of the new composed system from the entropy functions of its parts: this is where entropy maximization comes in.

For a nice introduction to these ideas, check out Owen’s blog article:

• Owen Lynch, Compositional thermostatics, Topos Institute Blog, 9 September 2021.

And then comes the really interesting part: checking that this adequately captures many of the examples physicists have thought about!

Here’s one: two cylinders of ideal gas with a movable divider between them that’s permeable to heat.

Yes, this is an operation in our operad—and if you tell us the entropy function of each cylinder of gas, our formalism will automatically compute the entropy function of the resulting combination of these two cylinders.

There are many other examples. Did you ever hear of the ‘canonical ensemble’, the ‘microcanonical ensemble’, or the ‘grand canonical ensemble’? Those are famous things in statistical mechanics. We show how our formalism recovers these.

I’m sure there’s much more to be done. But I feel happy to see modern math being put to good use: making the foundations of thermodynamics more precise. Once Vladimir Arnol’d wrote:

Every mathematician knows that it is impossible to understand any elementary course in thermodynamics.

I’m not sure our work will help with that — and indeed, it’s possible that once the mathematicians finally understand thermodynamics, physicists won’t understand what the mathematicians are talking about! But at least we’re clearly seeing some more of the mathematical structures that are hinted at, but not fully spelled out, in such an elementary course.

I expect that our work will interact nicely with Simon’s work on the Legendre transform. The Legendre transform of a concave (or convex) function is widely used in thermostatics, and Simon describes this for functions valued in $[-\infty,\infty]$ using enriched profunctors:

• Simon Willerton, Enrichment and the Legendre–Fenchel transform I, The n-Category Café, April 16, 2014.

• Simon Willerton, Enrichment and the Legendre–Fenchel transform II, The n-Category Café, May 22, 2014.

He also has a paper on this, and you can see him talk about it on YouTube.